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On stochastic multi-group Lotka-Volterra ecosystems with regime switching
A preconditioned fast Hermite finite element method for space-fractional diffusion equations
1. | School of Mathematics, Shandong University, Jinan, Shandong 250100, China |
2. | Department of Mathematics, University of South Carolina, Columbia, SC 29208, USA |
We develop a fast Hermite finite element method for a one-dimensional space-fractional diffusion equation, by proving that the stiffness matrix of the method can be expressed as a Toeplitz block matrix. Then a block circulant preconditioner is presented. Numerical results are presented to show the utility of the fast method.
References:
[1] |
T. Basu and H. Wang,
A fast second-order finite difference method for space-fractional diffusion equations, Int. J. Numer. Anal. Mod., 9 (2012), 658-666.
|
[2] |
D. Benson, S. W. Wheatcraft and M. M. Meerschaert,
The fractional-order governing equation of Lévy motion, Water Resour. Res., 36 (2000), 1413-1423.
doi: 10.1029/2000WR900032. |
[3] |
W. Bu, Y. Tang and J. Yang,
Galerkin finite element method for two-dimensional Riesz space fractional diffusion equations, J. Comput. Phys., 276 (2014), 26-38.
doi: 10.1016/j.jcp.2014.07.023. |
[4] |
R. H. Chan and M. K. Ng,
Conjugate gradient methods for Toeplitz systems, SIAM Review, 38 (1996), 427-482.
doi: 10.1137/S0036144594276474. |
[5] |
R. H. Chan,
Circulant preconditioners for Hermitian Toeplitz systems, SIAM Journal on Matrix Analysis and Applications, 10 (1989), 542-550.
doi: 10.1137/0610039. |
[6] |
R. H. Chan and X. Q. Jin,
A family of block preconditioners for block systems, SIAM journal on scientific and statistical computing, 13 (1992), 1218-1235.
doi: 10.1137/0913070. |
[7] |
P. J. Davis,
Circulant Matrices, Wiley-Intersciences, New York, 1979. |
[8] |
W. Deng,
Finite element method for the space and time fractional Fokker-Planck equation, SIAM Journal on Numerical Analysis, 47 (2008), 204-226.
doi: 10.1137/080714130. |
[9] |
K. Diethelm and A. D. Freed,
An efficient algorithm for the evaluation of convolution integrals, Computers and Mathematics with Applications, 51 (2006), 204-226.
doi: 10.1016/j.camwa.2005.07.010. |
[10] |
N. Du and H. Wang,
A fast finite element method for space-fractional dispersion equations on bounded domains in $\mathbb{R}^2$, SIAM J. Sci. Comput., 37 (2015), A1614-A1635.
doi: 10.1137/15M1007458. |
[11] |
V. J. Ervin and J. P. Roop,
Variational formulation for the stationary fractional advection dispersion equation, Numer. Methods for Partial Differential Equations, 22 (2005), 558-576.
doi: 10.1002/num.20112. |
[12] |
V. J. Ervin and J. P. Roop,
Variational Solution of Fractional Advection Dispersion Equations on Bounded Domains in $R^{d}$, Numer. Methods for Partial Differential Equations, 23 (2007), 256-281.
doi: 10.1002/num.20169. |
[13] |
R. M. Gray,
Toeplitz and circulant matrices: A review, Communications and Information Theory, 2 (2016), 155-239.
doi: 10.1561/0100000006. |
[14] |
J. Jia and H. Wang,
A preconditioned fast finite volume scheme for a fractional differential equation discretized on a locally refined composite mesh, J. Comput. Phys., 299 (2015), 842-862.
doi: 10.1016/j.jcp.2015.06.028. |
[15] |
Y. Jiang and X. Xu,
Multigrid methods for space fractional partial differential equations, J. Comput. Phys., 302 (2015), 374-392.
doi: 10.1016/j.jcp.2015.08.052. |
[16] |
S. L. Lei and H. W. Sun,
A circulant preconditioner for fractional diffusion equations, J. Comput. Phys., 242 (2013), 715-725.
doi: 10.1016/j.jcp.2013.02.025. |
[17] |
C. Li, Z. Zhao and Y. Q. Chen,
Numerical approximation of nonlinear fractional differential equations with subdiffusion and superdiffusion, Computers and Mathematics with Applications, 62 (2011), 855-875.
doi: 10.1016/j.camwa.2011.02.045. |
[18] |
X. Li and C. Xu,
The existence and uniqueness of the weak solution of the space-time fractional diffusion equation and a spectral method approximation, Commun. Comput. Phys., 8 (2010), 1016-1051.
doi: 10.4208/cicp.020709.221209a. |
[19] |
F. R. Lin, S. W. Yang and X. Q. Jin,
Preconditioned iterative methods for fractional diffusion equation, J. Comput. Phys., 256 (2014), 109-117.
doi: 10.1016/j.jcp.2013.07.040. |
[20] |
Y. Lin and C. Xu,
Finite difference/spectral approximations for the time-fractional diffusion equation, J. Comput. Phys., 225 (2007), 1533-1552.
doi: 10.1016/j.jcp.2007.02.001. |
[21] |
F. Liu, V. Anh and I. Turner,
Numerical solution of the space fractional Fokker-Planck equation, J. Comput. Appl. Math., 166 (2014), 209-219.
doi: 10.1016/j.cam.2003.09.028. |
[22] |
C.~W. Lv and C. Xu,
Improved error estimates of a finite difference/spectral method for time-fractional diffusion equations, Int. J. Numer. Anal. Mod., 12 (2015), 384-400.
|
[23] |
M. M. Meerschaert, H. P. Scheffler and C. Tadjeran,
Finite difference methods for two-dimensional fractional dispersion equation, J. Comput. Phys., 211 (2006), 249-261.
doi: 10.1016/j.jcp.2005.05.017. |
[24] |
M. M. Meerschaert and C. Tadjeran,
Finite difference approximations for fractional advection-dispersion flow equations, J. Comput. Appl. Math., 172 (2004), 65-77.
doi: 10.1016/j.cam.2004.01.033. |
[25] |
R. Metzler and J. Klafter,
The random walk's guide to anomalous diffusion: A fractional dynamics approach, Phys. Rep., 339 (2000), 1-77.
doi: 10.1016/S0370-1573(00)00070-3. |
[26] |
H. K. Pang and H. W. Sun,
Multigrid method for fractional diffusion equations, J. Comput. Phys., 231 (2012), 693-703.
doi: 10.1016/j.jcp.2011.10.005. |
[27] |
I. Podlubny,
Fractional Differential Equations, Academic Press, New York, 1999. |
[28] |
Y. Saad,
Iterative Methods for Sparse Linear Systems, 2nd ed. , SIAM, Rhode Island, 2003.
doi: 10.1137/1. 9780898718003. |
[29] |
H. G. Sun, W. Chen and Y. Q. Chen,
Fractional differential models for anomalous diffusion, Physica A: Statistical Mechanics and its Applications, 389 (2010), 2719-2724.
doi: 10.1016/j.physa.2010.02.030. |
[30] |
H. Tian, H. Wang and W. Q. Wang,
An efficient collocation method for a non-local diffusion model, Int. J. Numer. Anal. Mod., 10 (2013), 815-825.
|
[31] |
P. Vabishchevich,
Numerical solution of nonstationary problems for a convection and a space-fractional diffusion equation, Int. J. Numer. Anal. Mod., 13 (2016), 296-309.
|
[32] |
H. Wang and D. P. Yang,
Wellposedness of variable-coefficient conservative fractional elliptic differential equations, SIAM Journal on Numerical Analysis, 51 (2013), 1088-1107.
doi: 10.1137/120892295. |
[33] |
H. Wang, D. P. Yang and S. F. Zhu,
Accuracy of finite element methods for boundary-value problems of steady-state fractional diffusion equations, Journal of Scientific Computing, 70 (2017), 429-449.
doi: 10.1007/s10915-016-0196-7. |
[34] |
H. Wang and X. H. Zhang,
A high-accuracy preserving spectral Galerkin method for the Dirichlet boundary-value problem of variable-coefficient conservative fractional diffusion equations, J. Comput. Phys., 281 (2015), 67-81.
doi: 10.1016/j.jcp.2014.10.018. |
[35] |
H. Wang and T. Basu,
A fast finite difference method for two-dimensional space-fractional diffusion equations, SIAM J. Sci. Comput., 34 (2012), A2444-A2458.
doi: 10.1137/12086491X. |
[36] |
H. Wang and N. Du,
A superfast-preconditioned iterative method for steady-state space-fractional diffusion equations, J. Comput. Phys., 240 (2013), 49-57.
doi: 10.1016/j.jcp.2012.07.045. |
[37] |
H. Wang, K. Wang and T. Sircar,
A direct $O(N\log^2 N)$ finite difference method for fractional diffusion equations, J. Comput. Phys., 229 (2010), 8095-8104.
doi: 10.1016/j.jcp.2010.07.011. |
[38] |
L. L. Wei, Y. N. He and Y. Zhang,
Numerical analysis of the fractional seventh-order KdV equation using an implicit fully discrete local discontinuous Galerkin method, Int. J. Numer. Anal. Mod., 10 (2013), 430-444.
|
show all references
References:
[1] |
T. Basu and H. Wang,
A fast second-order finite difference method for space-fractional diffusion equations, Int. J. Numer. Anal. Mod., 9 (2012), 658-666.
|
[2] |
D. Benson, S. W. Wheatcraft and M. M. Meerschaert,
The fractional-order governing equation of Lévy motion, Water Resour. Res., 36 (2000), 1413-1423.
doi: 10.1029/2000WR900032. |
[3] |
W. Bu, Y. Tang and J. Yang,
Galerkin finite element method for two-dimensional Riesz space fractional diffusion equations, J. Comput. Phys., 276 (2014), 26-38.
doi: 10.1016/j.jcp.2014.07.023. |
[4] |
R. H. Chan and M. K. Ng,
Conjugate gradient methods for Toeplitz systems, SIAM Review, 38 (1996), 427-482.
doi: 10.1137/S0036144594276474. |
[5] |
R. H. Chan,
Circulant preconditioners for Hermitian Toeplitz systems, SIAM Journal on Matrix Analysis and Applications, 10 (1989), 542-550.
doi: 10.1137/0610039. |
[6] |
R. H. Chan and X. Q. Jin,
A family of block preconditioners for block systems, SIAM journal on scientific and statistical computing, 13 (1992), 1218-1235.
doi: 10.1137/0913070. |
[7] |
P. J. Davis,
Circulant Matrices, Wiley-Intersciences, New York, 1979. |
[8] |
W. Deng,
Finite element method for the space and time fractional Fokker-Planck equation, SIAM Journal on Numerical Analysis, 47 (2008), 204-226.
doi: 10.1137/080714130. |
[9] |
K. Diethelm and A. D. Freed,
An efficient algorithm for the evaluation of convolution integrals, Computers and Mathematics with Applications, 51 (2006), 204-226.
doi: 10.1016/j.camwa.2005.07.010. |
[10] |
N. Du and H. Wang,
A fast finite element method for space-fractional dispersion equations on bounded domains in $\mathbb{R}^2$, SIAM J. Sci. Comput., 37 (2015), A1614-A1635.
doi: 10.1137/15M1007458. |
[11] |
V. J. Ervin and J. P. Roop,
Variational formulation for the stationary fractional advection dispersion equation, Numer. Methods for Partial Differential Equations, 22 (2005), 558-576.
doi: 10.1002/num.20112. |
[12] |
V. J. Ervin and J. P. Roop,
Variational Solution of Fractional Advection Dispersion Equations on Bounded Domains in $R^{d}$, Numer. Methods for Partial Differential Equations, 23 (2007), 256-281.
doi: 10.1002/num.20169. |
[13] |
R. M. Gray,
Toeplitz and circulant matrices: A review, Communications and Information Theory, 2 (2016), 155-239.
doi: 10.1561/0100000006. |
[14] |
J. Jia and H. Wang,
A preconditioned fast finite volume scheme for a fractional differential equation discretized on a locally refined composite mesh, J. Comput. Phys., 299 (2015), 842-862.
doi: 10.1016/j.jcp.2015.06.028. |
[15] |
Y. Jiang and X. Xu,
Multigrid methods for space fractional partial differential equations, J. Comput. Phys., 302 (2015), 374-392.
doi: 10.1016/j.jcp.2015.08.052. |
[16] |
S. L. Lei and H. W. Sun,
A circulant preconditioner for fractional diffusion equations, J. Comput. Phys., 242 (2013), 715-725.
doi: 10.1016/j.jcp.2013.02.025. |
[17] |
C. Li, Z. Zhao and Y. Q. Chen,
Numerical approximation of nonlinear fractional differential equations with subdiffusion and superdiffusion, Computers and Mathematics with Applications, 62 (2011), 855-875.
doi: 10.1016/j.camwa.2011.02.045. |
[18] |
X. Li and C. Xu,
The existence and uniqueness of the weak solution of the space-time fractional diffusion equation and a spectral method approximation, Commun. Comput. Phys., 8 (2010), 1016-1051.
doi: 10.4208/cicp.020709.221209a. |
[19] |
F. R. Lin, S. W. Yang and X. Q. Jin,
Preconditioned iterative methods for fractional diffusion equation, J. Comput. Phys., 256 (2014), 109-117.
doi: 10.1016/j.jcp.2013.07.040. |
[20] |
Y. Lin and C. Xu,
Finite difference/spectral approximations for the time-fractional diffusion equation, J. Comput. Phys., 225 (2007), 1533-1552.
doi: 10.1016/j.jcp.2007.02.001. |
[21] |
F. Liu, V. Anh and I. Turner,
Numerical solution of the space fractional Fokker-Planck equation, J. Comput. Appl. Math., 166 (2014), 209-219.
doi: 10.1016/j.cam.2003.09.028. |
[22] |
C.~W. Lv and C. Xu,
Improved error estimates of a finite difference/spectral method for time-fractional diffusion equations, Int. J. Numer. Anal. Mod., 12 (2015), 384-400.
|
[23] |
M. M. Meerschaert, H. P. Scheffler and C. Tadjeran,
Finite difference methods for two-dimensional fractional dispersion equation, J. Comput. Phys., 211 (2006), 249-261.
doi: 10.1016/j.jcp.2005.05.017. |
[24] |
M. M. Meerschaert and C. Tadjeran,
Finite difference approximations for fractional advection-dispersion flow equations, J. Comput. Appl. Math., 172 (2004), 65-77.
doi: 10.1016/j.cam.2004.01.033. |
[25] |
R. Metzler and J. Klafter,
The random walk's guide to anomalous diffusion: A fractional dynamics approach, Phys. Rep., 339 (2000), 1-77.
doi: 10.1016/S0370-1573(00)00070-3. |
[26] |
H. K. Pang and H. W. Sun,
Multigrid method for fractional diffusion equations, J. Comput. Phys., 231 (2012), 693-703.
doi: 10.1016/j.jcp.2011.10.005. |
[27] |
I. Podlubny,
Fractional Differential Equations, Academic Press, New York, 1999. |
[28] |
Y. Saad,
Iterative Methods for Sparse Linear Systems, 2nd ed. , SIAM, Rhode Island, 2003.
doi: 10.1137/1. 9780898718003. |
[29] |
H. G. Sun, W. Chen and Y. Q. Chen,
Fractional differential models for anomalous diffusion, Physica A: Statistical Mechanics and its Applications, 389 (2010), 2719-2724.
doi: 10.1016/j.physa.2010.02.030. |
[30] |
H. Tian, H. Wang and W. Q. Wang,
An efficient collocation method for a non-local diffusion model, Int. J. Numer. Anal. Mod., 10 (2013), 815-825.
|
[31] |
P. Vabishchevich,
Numerical solution of nonstationary problems for a convection and a space-fractional diffusion equation, Int. J. Numer. Anal. Mod., 13 (2016), 296-309.
|
[32] |
H. Wang and D. P. Yang,
Wellposedness of variable-coefficient conservative fractional elliptic differential equations, SIAM Journal on Numerical Analysis, 51 (2013), 1088-1107.
doi: 10.1137/120892295. |
[33] |
H. Wang, D. P. Yang and S. F. Zhu,
Accuracy of finite element methods for boundary-value problems of steady-state fractional diffusion equations, Journal of Scientific Computing, 70 (2017), 429-449.
doi: 10.1007/s10915-016-0196-7. |
[34] |
H. Wang and X. H. Zhang,
A high-accuracy preserving spectral Galerkin method for the Dirichlet boundary-value problem of variable-coefficient conservative fractional diffusion equations, J. Comput. Phys., 281 (2015), 67-81.
doi: 10.1016/j.jcp.2014.10.018. |
[35] |
H. Wang and T. Basu,
A fast finite difference method for two-dimensional space-fractional diffusion equations, SIAM J. Sci. Comput., 34 (2012), A2444-A2458.
doi: 10.1137/12086491X. |
[36] |
H. Wang and N. Du,
A superfast-preconditioned iterative method for steady-state space-fractional diffusion equations, J. Comput. Phys., 240 (2013), 49-57.
doi: 10.1016/j.jcp.2012.07.045. |
[37] |
H. Wang, K. Wang and T. Sircar,
A direct $O(N\log^2 N)$ finite difference method for fractional diffusion equations, J. Comput. Phys., 229 (2010), 8095-8104.
doi: 10.1016/j.jcp.2010.07.011. |
[38] |
L. L. Wei, Y. N. He and Y. Zhang,
Numerical analysis of the fractional seventh-order KdV equation using an implicit fully discrete local discontinuous Galerkin method, Int. J. Numer. Anal. Mod., 10 (2013), 430-444.
|
h | | | |
| | | |
h | | | |
| | | |
h | DOF | | order | | order | |
0.5 | | 16 | | | ||
| 32 | | 3.7693 | | 3.2858 | |
| 64 | | 3.9982 | | 3.3758 | |
| 128 | 4.0582 | 3.4636 | |||
| 256 | 4.0849 | 3.4673 | |||
0.1 | | 16 | | | ||
| 32 | | 3.6643 | | 2.6352 | |
| 64 | | 3.8792 | | 2.8536 | |
| 128 | 3.9547 | 2.9752 | |||
| 256 | 3.9185 | 3.0311 | |||
0.9 | | 16 | | | ||
| 32 | | 3.7104 | | 3.5593 | |
| 64 | | 3.9213 | | 3.7705 | |
| 128 | 3.9515 | 3.8202 | |||
| 256 | 4.0001 | 3.9131 |
h | DOF | | order | | order | |
0.5 | | 16 | | | ||
| 32 | | 3.7693 | | 3.2858 | |
| 64 | | 3.9982 | | 3.3758 | |
| 128 | 4.0582 | 3.4636 | |||
| 256 | 4.0849 | 3.4673 | |||
0.1 | | 16 | | | ||
| 32 | | 3.6643 | | 2.6352 | |
| 64 | | 3.8792 | | 2.8536 | |
| 128 | 3.9547 | 2.9752 | |||
| 256 | 3.9185 | 3.0311 | |||
0.9 | | 16 | | | ||
| 32 | | 3.7104 | | 3.5593 | |
| 64 | | 3.9213 | | 3.7705 | |
| 128 | 3.9515 | 3.8202 | |||
| 256 | 4.0001 | 3.9131 |
Gauss | CGS | |||
h | CPU(s) | Itr. # | CPU(s) | Itr. # |
0.0916 | 0.2103 | 137 | ||
| 0.4865 | 1.5369 | 272 | |
| 3.5858 | 9.8734 | 415 | |
| 24.4774 | 62.7682 | 665 | |
| 186.9874 | 391.5595 | 899 | |
| 1485.1450 | 2731.0751 | 1676 | |
| out of memory | out of memory |
Gauss | CGS | |||
h | CPU(s) | Itr. # | CPU(s) | Itr. # |
0.0916 | 0.2103 | 137 | ||
| 0.4865 | 1.5369 | 272 | |
| 3.5858 | 9.8734 | 415 | |
| 24.4774 | 62.7682 | 665 | |
| 186.9874 | 391.5595 | 899 | |
| 1485.1450 | 2731.0751 | 1676 | |
| out of memory | out of memory |
FCGS | SFCGS | CFCGS | ||||
h | CPU(s) | Itr. # | CPU(s) | Itr. # | CPU(s) | Itr. # |
0.0832 | 137 | 0.0305 | 7 | 0.0341 | 9 | |
| 0.1809 | 272 | 0.0358 | 7 | 0.0419 | 10 |
| 0.2890 | 415 | 0.0503 | 7 | 0.0445 | 11 |
| 0.9907 | 665 | 0.0680 | 8 | 0.0653 | 12 |
| 2.4525 | 899 | 0.0932 | 8 | 0.1046 | 14 |
| 16.0752 | 1676 | 0.1536 | 9 | 0.2422 | 16 |
| 26.4751 | 6421 | 0.1914 | 9 | 0.5955 | 19 |
| N/A | | 0.6319 | 10 | 1.4107 | 22 |
FCGS | SFCGS | CFCGS | ||||
h | CPU(s) | Itr. # | CPU(s) | Itr. # | CPU(s) | Itr. # |
0.0832 | 137 | 0.0305 | 7 | 0.0341 | 9 | |
| 0.1809 | 272 | 0.0358 | 7 | 0.0419 | 10 |
| 0.2890 | 415 | 0.0503 | 7 | 0.0445 | 11 |
| 0.9907 | 665 | 0.0680 | 8 | 0.0653 | 12 |
| 2.4525 | 899 | 0.0932 | 8 | 0.1046 | 14 |
| 16.0752 | 1676 | 0.1536 | 9 | 0.2422 | 16 |
| 26.4751 | 6421 | 0.1914 | 9 | 0.5955 | 19 |
| N/A | | 0.6319 | 10 | 1.4107 | 22 |
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